Chapter 11.3, Problem 8E

Solution

(a)

To explain: Whether the model is biased or not if a sum of seven is observed rolling two dice four times in a row.

The occurrence of a sum of 7 when two dice are rolled four times in a row should not question the model, though the likelihood of the occurrence of a sum of 7 four times in a row is very low it can be considered a rare event.

Given:

Two dice are rolled four times in a row and a sum of 7 is observed.

Calculation:

The total number of ways sum of seven can occur when two dice are rolled is $6\left\{(1,6)(2,5)(3,4)(4,3)(5,2)(6,1)\right\}$ .

Total number of possibilities when two dice are rolled is $=6\times 6=36$ .The probability that a sum of 7 is observed when two dice are rolled is $=\frac{6}{36}=\frac{1}{6}$ .

The probability that a sum of seven is observed four times in a row $={\left(\frac{1}{6}\right)}^4=\frac{1}{1296}$ .

Interpretation:

The occurrence of the sum of seven four times in a row is 1 in 1296 trials. Even though the probability of the event occurring is very low, it can be a rare event and the model can be assumed to be normal. So there is no need to question the model.

(b)

To explain: Whether the model is biased or not if a sum of two is observed rolling two dice four times in a row.

The occurrence of a sum of 2 when two dice are rolled four times in a row should question the model, as the likelihood of the occurrence of a sum of 2 four times in a row is very low and not normal.

Given:

Two dice are rolled four times in a row and a sum of 2 is observed.

Calculation:

The total number of ways the sum of two can occur when two dice are rolled is $=1\left\{(1,1)\right\}$ .

Total number of possibilities when two dice are rolled is $=6\times 6=36$ .The probability that a sum of 2 is observed when two dice are rolled is $=\frac{1}{36}$ .

The probability that a sum of two is observed four times in a row $={\left(\frac{1}{36}\right)}^4=\frac{1}{1679616}$Interpretation:

The occurrence of the sum of two four times in a row is 1 in 1679616 trials. Thus it is not considered to be normal. As the probability of the event occurring is very low, the model has to be questioned and the model can be biased.